Considerable progress has been made in understanding elastic
depinning, in which we neglect the possibility of phase slip. A
classic model is simply to use a ``Langevin'' dynamics, e.g., for the
CDW phase,

where I have dropped the thermal noise present at . is just the continuum Hamiltonian in Eq. 172. This is
known as the Fukuyama-Lee-Rice equation.

A priori, at T=0, there is no guarantee of reaching a steady state
at long times. In purely elastic models of random manifolds and
random media, an exact result, due to Alan Middleton, tells us that
these models have a unique long-time steady state in the sliding
phase (of course, the pinned state is hysteretic even in elastic
models). This ``theorem'' breaks down once phase slips are allowed in
the model, and more complex behavior including multiple steady states
and chaotic dynamics is allowed.

One extremely simple model of CDW depinning can be easily solved,
which is in effect the case of d=0, or a single degree of freedom.
The equation of motion in this case is

where and . This is in
fact the equation describing the dynamics of an overdamped Josephson
junction in an external current. Then corresponds to the phase
difference between two superconductors across the junction. The force
F is proportional to the current, and the voltage .

It is clear that, for , the phase asymptotically reaches
a constant value, where

The threshold force, , occurs at the point at which this equation
can no longer be satisfied, i.e. . For , the
voltage is always non-zero. But we can solve Eq. 221 by
separation of variables

taking for simplicity as the initial condition. For
, the integral is dominated by the maxima of Y.
For long times (large ), there are approximately such maxima, each contributing equally to the integral
(since the integrand is periodic). Expanding Y near a maxima,
e.g. , we
have

The implicit solution is then

which implies

with . The exponent is one of several interesting
quantities to study more generally at a depinning transition.

Space does not permit me to discuss the full details of the theory of
depinning in finite dimensions. The interested reader is encouraged
to look at the references at the end of the notes. Instead, I will
present some of the scaling arguments that provide a physical picture
of depinning. For variety, I do this here for the case of a driven
domain wall[11]. Very similar considerations apply to
CDWs, and are discussed in depth in Refs.[15]. We can
imagine field-cooling a dirty ferromagnet to arrive at a well-ordered
state. After turning off the field, we can force the introduction of
a domain wall by imposing opposite magnetizations on two ends of the
sample. Applying a field at this point imposes a force on the domain
wall in the direction that increases the magnetization along the
field.

At zero temperature, where no thermal activation is possible, the
domain may be stuck in a local minima of the random potential. For
small applied forces, only transient motion will result, ending in an
stationary configuration for the domain with asymptotically zero
velocity. If enough force is applied to overcome all the local
pinning forces, the domain will slide with a non-zero mean velocity.
Somewhere in between there must be a depinning point, at which the
domain's mean velocity goes to zero. We would like to understand as
much as possible the approach to this point from above and below, as
well as the general behavior of the system in both the pinned and
moving ``states''.

where is a kinetic (drag) coefficient and F is an external
force. Using the form of the Hamiltonian, we have

where , , and is the quenched random local force. In fact, a systematic
analytical treatment can be made for this model, and I encourage you
to look at the relevant references. The approach is technically
similar to the functional RG approach just discussed for equilibrium
systems, though the physics is in many ways quite different. Here I
will just discuss the behavior in a phenomenological way and describe
the results we can obtain just from scaling.

Let us first consider the behavior at low forces. As the force is
increased from zero, the interface will slide up against the potential
barriers around the initial state, with only small smooth
rearrangements of the configuration. Eventually, however, the applied
force will be sufficient to overcome some local pinning force and that
region will jump forward into a new metastable configuration. At low
forces, this events will be widely separated and typically small.
There will, of course, be such events at every force in an infinite
sample, but they will be quite far apart. Likewise, certain rare
regions of the sample may have anomalously low pinning forces, and
these regions will enable very rare large jumps even at low forces.
We expect that the ``rare'' events will actually be exponentially
unlikely, since, for instance, the probability of finding an area of
linear size L with anomalously low pinning forces is roughly

where a is some correlation length for the random potential, and
.

Both large and small events may be thought of as avalanches. As we
increase the force f, the typical size of avalanches will increase,
as will the distance between avalanches. Another effect reinforcing
this trend is that smaller avalanches will essentially trigger
neighboring areas to jump as well. As we increase f close to ,
the critical or threshold force, even the typical avalanche size
becomes much larger than the correlation length for the disorder.
Then we can expect a scaling form for the distribution of avalanche
size induced upon increasing the force an infinitesimal amount. Let's
denote the probability (per unit volume) of finding an avalanche of
size larger than as

where this form applies only for , and is the
typical avalanche size. We expect that the scaling function

for large x.

We should also note that these avalanches will typically begin and end
in rough configurations of the interface. They can thus be
characterized by self-affine scaling, just as in equilibrium. So we
should define an ``avalanche roughness'' exponent,

for . Because this roughness is only really defined in
the scaling limit where and diverge, it is really a
property of the threshold system. The divergence of should also
be characterized by a scaling law,

Large avalanches likewise require a long time to move. As we approach
threshold from below, any small change of the force thus causes a
rather long-lived disturbance. The lifetime of these jumps may be
denoted

defining a conventional dynamical critical exponent z.

We seem to have developed a profusion of unknown exponents! Let us
take stock for a moment and attempt to derive some relations between
them. To do so, consider the response of the interface to an
infinitesimal perturbation. Let us imagine adding a small external
force to the equation of motion

We may then define a sort of susceptibility for the interface as

We can use the rotational invariance of the system to constrain
. To do so, let us specialize to the case where is
independent of time. Then we can make the change of variables

This removes from the right-hand-side of Eq. 228, but
it re-appears inside the random force

where FT indicates a Fourier transform. As in the equilibrium
problem, however, the distribution of is unaffected by this
shift, so the average properties of the shifted equation are as if
. Thus the only contribution to comes from the
constant shift of u, and we have

We thus expect to have the general scaling form

If we now first take , then , we expect to get a finite result, and we have

However, in physical terms, we know that

Equating the two requires

We can also determine the avalanche exponent . Again, this
mean susceptibility is the mean change in u as f is increased.
For an avalanche of size , this change is if the
avalanche contains the point of interest. For a volume of size L,
the probability density of such avalanches is

The mean change in u at a point in this volume is then

Integrating this over gives

Doing this integral and comparing gives .

This gives rather a complete picture below threshold. What happens
above threshold? Well, as , the avalanches become
larger and larger and eventually one giant avalanche brings the system
into motion at . At that point, the system begins to slide.
We may estimate the velocity above threshold by

Near threshold, the motion is ``jerky'', with large regions sticking
for long times. We may reinterpret the correlation lengths and times
at these sticking scales and times on this side of the phase
transition.

It only remains to determine the two unknown exponents, say z and
. These require an analytic field-theoretical RG treatment
to obtain. The results are, however,